This question has been on my mind for a very long time, and I thought I'd finally ask it here.

When I was 6, my dad pulled me out of school. The classes were too easy; the professors, too dull. My father had been man of philosophy his entire life (almost got a PhD in it) and regretted not having a more quantitive background. He wanted me to have a different life and taught me math accordingly. When I was 11, I taught myself trig. When I was 12, I started taking calculus at my local university. I continued on this track, and finally got to real analysis and abstract algebra at 15. I loved every math course I ever took and found myself breezing through all that was presented to me (the university was not Princeton after all). However, around this time, I came to the conclusion that math was not for me. I decided to try a different path.

Why, you might ask, did I do this? The answer was simple: I didn't believe I could be a great mathematician. While I thrived taking the courses, I never turned math into a lifestyle. I didn't come home and do complex questions on a white board. I didn't read about Euler in my spare time. I also never felt I had a great intuition into problems. Once you showed me how to solve a problem, I was golden. But start from scratch on my own? It seemed like a different story entirely. To make things worse, my sister, who was at Caltech at the time, would call home with stories of all these incredible undergrads who solved the mathematical mysteries of the universe as a hobby. Whenever I mentioned math as a career, she would always issue a strong warning: you're not like these kids who spend all their time doing math. Think about doing something else.

Over time, I came to agree with this statement. Coincidentally, I got rejected by MIT and Princeton to continue my undergraduate studies there. This crushed me at the time; my dream of studying math at one of the great institutions had ended. Instead, I ended up at Georgia Tech (not terrible by any means, just not what I had envisioned). Being at an engineering school, I thought I'd give aerospace a shot. It had lots of math, right? Not really, or at least not enough for my taste. I went into CS. This was much better, but still didn't feel quite right. At last, as a sophomore, I felt it was time to get back on track: I'm now doubling majoring in applied math and CS.

My question is, how do I know I'm not making a mistake? There seems to be so many people doing math competitions, research, independent studies, etc, while I just started to take some math courses again. What should I do to test myself and see if I can really make math a career? I apologize for the long and possibly quite subjective post. I'd just really like to hear from math people who know their stuff. Thanks a bunch in advance.

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

$\begingroup$The primary reason that mathematicians (at least the great one) doing math is they find doing math is fun. If you find doing math is fun, then go ahead. If not, then it didn't hurt to learn some math and apply it to whatever field you will be in. However, it won't be a good choice/direction for your future career.$\endgroup$
– achille huiSep 11 '14 at 6:03

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$\begingroup$Strive not to be great amongst everyone, but great amongst yourself. The first thing will be decided and worked out by everyone else. In other words, don't worry so much about how awesome you are. The field is full of not-even-remotely great researchers , and they can each be important and happy.$\endgroup$
– zibadawa timmySep 11 '14 at 6:30

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$\begingroup$This question needs to be more quantitative. How about, "How can I find out my utility function?"$\endgroup$
– KeenSep 11 '14 at 14:23

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$\begingroup$Just remember that the remorse of not following your interest is a slow and killing pain, don't ever let it come to you. Do what you find interesting, when you find interesting. Cheers!$\endgroup$
– user3459110Sep 11 '14 at 17:09

12 Answers
12

I don't think you can "know" that you aren't making a mistake. This applies to almost everything in life. What I would say is that if you want to do something, then do it. Life is too short to let worries paralyse you into doing nothing. Get into the course, focus your study on the parts that are most interesting, and see where you are at the end of the course. The worst that will happen is that you end up with a good degree and a good recreational mathematics habit (unless you drop out to follow your heart elsewhere).

I am absolutely certain that I am not made to push the boundaries of mathematics, but having a degree in the subject has opened many doors for me. And now that I'm spending my days at home, looking after my son, it gives me the ability to keep my brain active by studying alone at home.

There is a lot of hype surrounding "math competitions", since "winning" and "competition" can easily be understood in broad cultural terms, whether or not we collectively think of these as either highest goals or as legitimate formative principles. Most professional mathematics and its practice does not resemble competition-math at all, specifically, serious projects have substantial background requirements, may take months or years to complete or partially complete, and have meaning beyond "getting the result before anyone else" (although some people've been so indoctrinated in math-as-game that they never recover, and find no other meaning in it).

Also, there is much hype about "the other people" being incredible geniuses, while few sane people will look at themselves and see "a genius". :) But this is mostly gossip or mythology, in part generated to create a superficial excitement where otherwise there'd be mostly hard work without pop-style-glamour. :) And, of course, there is a common style of "bluffing" and/or never admitting weakness or ignorance, but this is mostly a facade, whose maintenance is most enthusiastic among those most worried about the "game" aspect of mathematics or anything else. Such stuff should not be unquestioningly believed.

To make a living as a mathematician, and to make reasonable contributions, it is not necessary to be a larger-than-life romantic-heroic figure. :) Possibly many of us, occasionally, wish that we were such a figure, but that more comic-book or video-game reality than any human-lifetime reality.

More important than hype is the concrete reality of how one spends one's days: if you like thinking about mathematical things, perhaps teaching mathematical things, then being a mathematician is a happy job, with or without rich-and-famousness. The purported glamor of "being a great [whatever]" is not a reliable thing to aim toward, since the "process" of practicing the [whatever] is how one will spend one's days. It is a reasonable analogy, I think, to say that "great musicians" occur by accident among the class of people who really, really enjoy practicing and "jamming" (in whatever genres), rather than people who wish for celebrity but hate practicing. Of course there're the people who pose as never practicing, but reality belies that cute P.R. pose...

You shouldn't judge yourself or your achievements by other people's standards. Do mathematics because you enjoy it, and find it interesting. Many great discoveries have come from average minds in a spirit of idle curiosity and playfulness. Likewise many brilliant minds achieve very little. In research, your approach matters as much as your intellect.

In short, you appear to be suffering performance anxiety because of your own or others' expectations. Get rid of those expectations and you will be free to express your potential, whatever that may be.

I loved math in school but struggled at uni, and lost plenty of motivation when unable to connect what I was doing to any sort of career (pure "theoretical" math was my strength). It did however lead into cryptography, programming and IT which is now the career I love.

What I'm attempting to express is that you can't know if it's a mistake. I could have made 2 extra years of a good income if I had chosen the right degree initially, but I owe discovering my real passion to attempting one that failed.

One thing I learned as a pure math enthusiast:
Pick the odd one out:
1. A phd in applied physics
2. A phd in pure math
3. A large pizza
4. A phd in financial math

As the mother of a high school junior taking all AP math/physics because he wants to be able to choose whatever path he determines is right for him, I wish more teachers were passionate about teaching. It would make a world of difference if a teacher would communicate HOW math and physics are present in our world, getting kids interested and curious by engaging them. The AP Physics teacher puts problems on the board and tells the students to solve them. How about showing a clip from an action movie and then discussing how physics is at work, or something similar.Stormwater problems, building design, all could used to illustrate the beauty of physics, and maybe lead to interest in innovation or solving problems.

I interpret your question as how to determine
if you are good enough to be a mathematics professor
and to do academic research for a living.

Note: You may wish to ask your question on http://academia.stackexchange.com
as there are quite a few mathematician professors there
who would be able to give you their perspective.

My background:
My father has a PhD in pure math.
I had the privilege of taking part in the International Math Olympiad.
I wanted to be a mathematician just like my father,
but because there are so few tenure-track professor jobs in pure math,
he encouraged me to study something more applied but still math-y.
I ended up doing a double degree in CS and math,
just like the OP.
For graduate school, I got a PhD in operations research,
which is basically applied math
(we apply optimization and probability/statistics to real-life problems).

Question: How did I know if I am good enough to be an academic researcher?

Technically correct but useless answer:
When I have published lots of papers
and gotten tenure at a good university,
then I know that I am good enough to be an academic researcher.

More useful answer:

You take increasingly advanced math courses
including graduate-level math courses.

If you have a chance to do a UROP,
then do it to see what research math is like,
which is quite different from solving problem sets.
Also it is useful in getting a recommendation from a professor
for your application to graduate school.

As you get more advanced in your studies,
you will discover how good you are at math.

The reality is that it is very hard to get a job as a math professor
because there is much more supply (math PhDs) than demand (jobs).
To be a math professor is like being a professional athlete or musician
in that you have to be in the 0.0001% of your field.
Don't feel too disappointed if you can't make it,
along the way you may find something else that is challenging and that you like.

When you apply to math departments for graduate school,
if you are not good enough (in their eyes) to be a academic math researcher,
they will communicate that to you through their rejection.
That may be good news because you can try to do something else that is math-y
but less difficult to get into (e.g. CS, OR, statistics, financial math etc)
and your skills wouldn't have been wasted.

If you are accepted into a math graduate program, great!
Work hard on research.
However, due to the low probability of success at becoming a math professor,
I feel that it is wise to have an exit strategy;
i.e. develop skills such as programming
that will you to find job outside of math academia.

Computer Science with Math is a wonderful combination that opens up a lot of opportunities. You could very easily do something for either path and use a fair bit from both along the way.

I had a similar experience with math growing up, though not quite to the degree that you expressed. I don't think I am one of the "enlightened few" of Mathematics either, but I enjoyed it and thus stuck with it. I graduated from University earlier this year with a double major in Computer Science and Mathematics.

I am presently employed doing mostly Computer Science related activities, though I still find that my math background has strengthened my problem solving abilities. Math can be very valuable to your understanding of Data Structures and Algorithms.

I would say that if you enjoy both of your areas of study, your current path will open up many career possibilities.

I was a bit like you, always excelling at math in school, taking extra courses etc. but I am exactly the opposite of you. I love solving challenges but fail doing things 'like in the handbook' and so I found computer science for myself. There I can solve problems in many different ways and challenge my mind while still having enough math.

Like yu a looked at all the other people my age, winning prizes, doing stuff in their spare time and simply being better than me. But with some time I found what all the other good people here already told you: "There will always be someone better than you, but that shouldn't hinder you from doing wat you want to do! And makes your work no less important!"

While we humans always see the 1% standing out and feel bad, we forget 99% of all others struggling like us. And even being one of the 1% we will only see the 1% in any other part in life.

Good grades and an university with good reputations are fine. But they don't define you and what you want. Just look forward and ask yourself at every junction "where do I want to go NOW?" and never regret that decision. It is a part of you and if you come to the next junction in life you will be wiser than before.

Your current situation is somewhat similar to what occurred during my adolescent years. I grew up in a rather unstable and meager lifestyle; I never had any opportunities to gain a good education. From what I can remember, I earned first place in a district mathematics competition during the fourth grade, though I did not enjoy mathematics at that time. Regardless, I began studying trigonometry when I was thirteen, having seen a sine wave somewhere. I was enthralled by trigonometry! I often spent hours reading on the subject and began to marvel at how intriguing it all was. Afterwards, I bought a book on the calculus from money that I had gathered; over the course of the next few months, I learned both single variable and multi-variable calculus. I never had any joys in the years before that, but studying the calculus made me smile for once. It was the only thing that kept me going, I believe.

But then one day in secondary school I realised that no one cared. I had learned more mathematics than was taught in the school curriculum, yet no one cared. I went to the teachers and the advisers about my situation, and while their intentions were pure, they could not help me. "Testing out of" the various mathematics courses costed a lot of money, which I did not have. Enrolling in courses at the local junior college required prerequisites and more money, which I did not have. It was at this point that I further realised that pursing mathematics was a waste of time, as no one cared. In a sense, I began to not care.

It is here that are situations differ. You are uncertain as to your ability to compete with others in mathematics, while my predicament centered upon the fact that no one cared about my ability. As such, I must ask you one question: What do you see when you look upon any mathematics whatsoever? I used to see wonderful ideas and concepts, but I now only see mere symbols, nothing more. The spark inside me is gone I am afraid and so I will never pursue mathematics any more than evaluating an integral here and there for the intellectual challenge. If you look upon mathematics and see something grand, then I believe that you should pursue such regardless if you do not prove a famous theorem.

As for me, I decided to take the opposite road and did not study mathematics in a formal setting, such as a university. With great disagreement to my previous character, I have found a fondness of the arts and writing. When I am not programming, I spend my time writing great stories of things that I will never do or see. I would not have it any other way.

It doesn't really matter if you decide your career while in undergrad. You're there to develop a strong baseline and to learn about what you don't know yet. Honestly the one piece of advise I give to most undergrads is to take more math courses, even if it's unrelated to their field of study, because math will open doors that most people won't venture down. You don't need to be a mathematician to major in math. I majored in math and marine biology and it was the best decision because it set me apart from other "pure science" undergrads when I was applying for grad school. So you don't need to know if you're going to "become a mathematician" because your goal right now should be to absorb as much information as you can so that you can apply it to the field you end up in, be it math or something else.

This would be my personal answer. I have medals in international physics olympiad. What do I do now?

I am a businessman.

Yes. I love Math. I love money and women more.

No I will not be a great mathematician or physicist. Why would I?

I still love Math. I read calculus, accounting. But then again, besides programming skills, most truly advance Math is useless in real life. Eventually it's money that turns me on.

People say do what you love rather than make money. That's true. But keep an eye on money.

As of now, I really don't feel like learning more Math. I want to learn more about biz, having the right mindset, law of attraction, evolutionary psychology, and so many things that are actually useful. Yea I can derive most college level and high school level formula. Why invent? Just learn makes me happy. Why discover new things for others?

I am still able to derive all Math formulas all the way to colleges I think on top of my head. I taught Math when I was poor and I love it. But then, I have no time for it. In a sense I still love Math. Being a businessman means I no longer have time to learn it. I wish I can pass on my skills in learning to others but alas, I'll die with it too I guess.

I once use my math skills to taught high school children Math. They are impressed that I can derive all those formulas. But that's it. A $100 a month job. Needless to say I made far more money in biz. I honestly do not think people like me are meant to get a job anyway. So yea, that's how useful Math is. Interpersonal skills? Applied Math? Programming? Internet marketing? Now that gets me very far.

If anyone wants to verify my IPHO claim privately and know how to do so, let me know. Really. It's my life skills learned through experiences, rather than Math is how I got where I am. It's what gets me cash. It's what gets me laid. It's what gets me a wife (that I am now divorcing because my math shows that marriage is inefficient).

$\begingroup$Thio is not my official name. I only tell this privately and we can't do pm here.$\endgroup$
– user4951Sep 11 '14 at 16:08

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$\begingroup$"I still love Math...As of now, I really don't feel like learning more Math." I would ask you what your definition of "love" is based on those contradictory claims, but reading your thoughts on women and money already give insight.$\endgroup$
– daOnlyBGOct 19 '14 at 21:27

$\begingroup$I am still able to derive all Math formulas all the way to colleges I think on top of my head. I taught Math when I was poor and I love it. But then, I have no time for it. In a sense I still love Math. Being a businessman means I no longer have time to learn it. I wish I can pass on my skills in learning to others but alas, I'll die with it too I guess.$\endgroup$
– user4951Oct 21 '14 at 12:35